OFFSET
0,8
COMMENTS
T is the convolution triangle based on T(n,0) = A004148(n-1) (n >= 1). - Peter Luschny, Oct 19 2022
FORMULA
EXAMPLE
T(5,2)=3 because we have (UD)(UD)UUDUDD, (UD)UUDUDD(UD), and UUDUDD(UD)(UD) (the hills are placed between parentheses).
Triangle starts:
1;
0, 1;
1, 0, 1;
1, 2, 0, 1;
2, 2, 3, 0, 1;
4, 5, 3, 4, 0, 1;
MAPLE
g := ((1-z-z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^3: G := 1/(1-t*z-z^2-z^3*g): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
# Alternative based on a modified form of A004148:
# Uses function PMatrix from A357368. Adds column 1, 0, 0, 0, ... to the left.
M004148 := n -> `if`(n<3, 2-n, hypergeom([(2-n)/2, (3-n)/2, (3-n)/2, (4-n)/2], [2, 2-n, 3-n], 16)):
PMatrix(10, n -> simplify(M004148(n))); # Peter Luschny, Oct 19 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Oct 11 2009
STATUS
approved